# Properties

 Label 4025.2.a.r Level $4025$ Weight $2$ Character orbit 4025.a Self dual yes Analytic conductor $32.140$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.122821.1 Defining polynomial: $$x^{5} - 2x^{4} - 4x^{3} + 4x^{2} + 3x - 1$$ x^5 - 2*x^4 - 4*x^3 + 4*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 805) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{4} + \beta_{3}) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{6} + q^{7} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{8} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b3 + 1) * q^3 + (b4 + b3) * q^4 + (b3 - b2 - 2*b1 + 2) * q^6 + q^7 + (b4 + b3 - b2) * q^8 + (2*b3 - b2 - b1 + 1) * q^9 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{4} + \beta_{3}) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{6} + q^{7} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{8} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{11} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{12} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{13} + ( - \beta_1 + 1) q^{14} + ( - \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 1) q^{16} + ( - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 2) q^{17} + (\beta_{4} + 4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{18} + ( - \beta_{4} - 3 \beta_1 + 2) q^{19} + (\beta_{3} + 1) q^{21} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{22} - q^{23} + (3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{24} + (\beta_{4} - \beta_{3} - 4 \beta_1 + 3) q^{26} + (\beta_{4} + \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 4) q^{27} + (\beta_{4} + \beta_{3}) q^{28} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2) q^{29} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{31} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{32} + ( - 2 \beta_{4} - 5 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{33} + ( - \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{34} + (3 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 5) q^{36} + ( - 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{37} + (2 \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 4) q^{38} + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{39} + (2 \beta_{4} + 4 \beta_1 - 1) q^{41} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{42} + ( - \beta_{4} + \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 1) q^{43} + ( - 3 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + \beta_1) q^{44} + (\beta_1 - 1) q^{46} + ( - 2 \beta_{4} - \beta_{2} - \beta_1 + 6) q^{47} + (2 \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{48} + q^{49} + ( - 4 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{51} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 7) q^{52} + (2 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{53} + (5 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 7) q^{54} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{56} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 6) q^{57} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{58} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{59} + (3 \beta_{2} + \beta_1 + 1) q^{61} + (3 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{62} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{63} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 3) q^{64} + ( - 3 \beta_{4} - 7 \beta_{3} + 6 \beta_{2} + 9 \beta_1 - 10) q^{66} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 6) q^{67} + ( - \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + \beta_1 - 8) q^{68} + ( - \beta_{3} - 1) q^{69} + (2 \beta_{4} + 4 \beta_{3} + \beta_{2} + 5 \beta_1 - 10) q^{71} + (5 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 5 \beta_1 + 6) q^{72} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{73} + ( - 4 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{74} + (2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 3) q^{76} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{77} + (2 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} - 7 \beta_1 + 3) q^{78} + (2 \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_1 + 1) q^{79} + (5 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 4) q^{81} + ( - 2 \beta_{4} - 4 \beta_{3} - 5 \beta_1 - 3) q^{82} + ( - \beta_{4} - \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 2) q^{83} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{84} + ( - 5 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 5 \beta_1) q^{86} + ( - 7 \beta_{4} - 4 \beta_{3} + \beta_{2} - 2 \beta_1 + 8) q^{87} + ( - 4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 7) q^{88} + ( - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2}) q^{89} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{91} + ( - \beta_{4} - \beta_{3}) q^{92} + (3 \beta_{4} + \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 4) q^{93} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{94} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{96} + (5 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{97} + ( - \beta_1 + 1) q^{98} + ( - 5 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 10) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b3 + 1) * q^3 + (b4 + b3) * q^4 + (b3 - b2 - 2*b1 + 2) * q^6 + q^7 + (b4 + b3 - b2) * q^8 + (2*b3 - b2 - b1 + 1) * q^9 + (-b3 + 2*b2 - 2) * q^11 + (2*b4 + 2*b3 - 2*b2 - b1 + 2) * q^12 + (2*b4 + b3 + b2 + b1) * q^13 + (-b1 + 1) * q^14 + (-b4 - 2*b2 - 2*b1 + 1) * q^16 + (-2*b4 - 3*b3 - 2*b1 + 2) * q^17 + (b4 + 4*b3 - 3*b2 - 2*b1 + 3) * q^18 + (-b4 - 3*b1 + 2) * q^19 + (b3 + 1) * q^21 + (-3*b3 + 3*b2 + 3*b1 - 1) * q^22 - q^23 + (3*b4 + 3*b3 - 2*b2 - b1 + 1) * q^24 + (b4 - b3 - 4*b1 + 3) * q^26 + (b4 + b3 - 3*b2 - 4*b1 + 4) * q^27 + (b4 + b3) * q^28 + (-3*b4 + 2*b3 + b2 - 2) * q^29 + (2*b4 + 2*b3 + b2 - b1 - 2) * q^31 + (-b4 + 2*b3 + 2*b1) * q^32 + (-2*b4 - 5*b3 + b2 + b1 - 3) * q^33 + (-b3 + 3*b2 + 5*b1 - 1) * q^34 + (3*b4 + 5*b3 - 5*b2 - 4*b1 + 5) * q^36 + (-2*b4 + b3 + 4*b2 + 2*b1) * q^37 + (2*b4 + 3*b3 + 2*b1 + 4) * q^38 + (3*b4 + 2*b3 - 2*b2 + b1 + 1) * q^39 + (2*b4 + 4*b1 - 1) * q^41 + (b3 - b2 - 2*b1 + 2) * q^42 + (-b4 + b3 + 3*b2 + 4*b1 + 1) * q^43 + (-3*b4 - 7*b3 + 2*b2 + b1) * q^44 + (b1 - 1) * q^46 + (-2*b4 - b2 - b1 + 6) * q^47 + (2*b3 - b2 - 4*b1 + 2) * q^48 + q^49 + (-4*b4 - 3*b3 + 3*b2 - b1 - 3) * q^51 + (b4 + b3 - b2 - b1 + 7) * q^52 + (2*b4 + b2 + b1 + 3) * q^53 + (5*b4 + 8*b3 - 4*b2 - 2*b1 + 7) * q^54 + (b4 + b3 - b2) * q^56 + (-2*b4 + b3 - 2*b2 - 6*b1 + 6) * q^57 + (-3*b4 + b3 - b2 + 3*b1 - 2) * q^58 + (-b4 - b3 - 2*b2 + b1) * q^59 + (3*b2 + b1 + 1) * q^61 + (3*b4 + 2*b3 - b2 - b1 + 4) * q^62 + (2*b3 - b2 - b1 + 1) * q^63 + (-b4 + 2*b2 + b1 - 3) * q^64 + (-3*b4 - 7*b3 + 6*b2 + 9*b1 - 10) * q^66 + (b4 + b3 + 2*b2 + b1 + 6) * q^67 + (-b4 - 3*b3 + 4*b2 + b1 - 8) * q^68 + (-b3 - 1) * q^69 + (2*b4 + 4*b3 + b2 + 5*b1 - 10) * q^71 + (5*b4 + 6*b3 - 4*b2 - 5*b1 + 6) * q^72 + (-b4 - 2*b3 + 2*b2 + 3*b1 + 1) * q^73 + (-4*b4 - 5*b3 + 3*b2 - b1 + 1) * q^74 + (2*b4 + b3 - 3*b2 - 5*b1 + 3) * q^76 + (-b3 + 2*b2 - 2) * q^77 + (2*b4 + 3*b3 - 4*b2 - 7*b1 + 3) * q^78 + (2*b4 - b3 + b2 + 5*b1 + 1) * q^79 + (5*b4 + 3*b3 - 3*b2 - 6*b1 + 4) * q^81 + (-2*b4 - 4*b3 - 5*b1 - 3) * q^82 + (-b4 - b3 + 4*b2 - 3*b1 + 2) * q^83 + (2*b4 + 2*b3 - 2*b2 - b1 + 2) * q^84 + (-5*b4 - 6*b3 + 2*b2 - 5*b1) * q^86 + (-7*b4 - 4*b3 + b2 - 2*b1 + 8) * q^87 + (-4*b4 - 4*b3 + 3*b2 + 3*b1 - 7) * q^88 + (-2*b4 + 5*b3 - 2*b2) * q^89 + (2*b4 + b3 + b2 + b1) * q^91 + (-b4 - b3) * q^92 + (3*b4 + b3 - 5*b2 - 4*b1 + 4) * q^93 + (-b4 + 2*b3 - b2 - 3*b1 + 4) * q^94 + (-2*b4 + b3 + b2 + 2*b1 + 5) * q^96 + (5*b4 - 2*b3 + b2 - 2*b1 + 1) * q^97 + (-b1 + 1) * q^98 + (-5*b4 - 8*b3 + 2*b2 + 7*b1 - 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 6 q^{3} + 3 q^{4} + 7 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10})$$ 5 * q + 3 * q^2 + 6 * q^3 + 3 * q^4 + 7 * q^6 + 5 * q^7 + 3 * q^8 + 5 * q^9 $$5 q + 3 q^{2} + 6 q^{3} + 3 q^{4} + 7 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9} - 11 q^{11} + 14 q^{12} + 7 q^{13} + 3 q^{14} - q^{16} - q^{17} + 17 q^{18} + 2 q^{19} + 6 q^{21} - 2 q^{22} - 5 q^{23} + 12 q^{24} + 8 q^{26} + 15 q^{27} + 3 q^{28} - 14 q^{29} - 6 q^{31} + 4 q^{32} - 22 q^{33} + 4 q^{34} + 28 q^{36} + q^{37} + 31 q^{38} + 15 q^{39} + 7 q^{41} + 7 q^{42} + 12 q^{43} - 11 q^{44} - 3 q^{46} + 24 q^{47} + 4 q^{48} + 5 q^{49} - 28 q^{51} + 36 q^{52} + 21 q^{53} + 49 q^{54} + 3 q^{56} + 15 q^{57} - 9 q^{58} - q^{59} + 7 q^{61} + 26 q^{62} + 5 q^{63} - 15 q^{64} - 45 q^{66} + 35 q^{67} - 43 q^{68} - 6 q^{69} - 32 q^{71} + 36 q^{72} + 7 q^{73} - 10 q^{74} + 10 q^{76} - 11 q^{77} + 8 q^{78} + 18 q^{79} + 21 q^{81} - 33 q^{82} + q^{83} + 14 q^{84} - 26 q^{86} + 18 q^{87} - 41 q^{88} + q^{89} + 7 q^{91} - 3 q^{92} + 19 q^{93} + 14 q^{94} + 26 q^{96} + 9 q^{97} + 3 q^{98} - 54 q^{99}+O(q^{100})$$ 5 * q + 3 * q^2 + 6 * q^3 + 3 * q^4 + 7 * q^6 + 5 * q^7 + 3 * q^8 + 5 * q^9 - 11 * q^11 + 14 * q^12 + 7 * q^13 + 3 * q^14 - q^16 - q^17 + 17 * q^18 + 2 * q^19 + 6 * q^21 - 2 * q^22 - 5 * q^23 + 12 * q^24 + 8 * q^26 + 15 * q^27 + 3 * q^28 - 14 * q^29 - 6 * q^31 + 4 * q^32 - 22 * q^33 + 4 * q^34 + 28 * q^36 + q^37 + 31 * q^38 + 15 * q^39 + 7 * q^41 + 7 * q^42 + 12 * q^43 - 11 * q^44 - 3 * q^46 + 24 * q^47 + 4 * q^48 + 5 * q^49 - 28 * q^51 + 36 * q^52 + 21 * q^53 + 49 * q^54 + 3 * q^56 + 15 * q^57 - 9 * q^58 - q^59 + 7 * q^61 + 26 * q^62 + 5 * q^63 - 15 * q^64 - 45 * q^66 + 35 * q^67 - 43 * q^68 - 6 * q^69 - 32 * q^71 + 36 * q^72 + 7 * q^73 - 10 * q^74 + 10 * q^76 - 11 * q^77 + 8 * q^78 + 18 * q^79 + 21 * q^81 - 33 * q^82 + q^83 + 14 * q^84 - 26 * q^86 + 18 * q^87 - 41 * q^88 + q^89 + 7 * q^91 - 3 * q^92 + 19 * q^93 + 14 * q^94 + 26 * q^96 + 9 * q^97 + 3 * q^98 - 54 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 4x^{3} + 4x^{2} + 3x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 2$$ v^3 - 2*v^2 - 3*v + 2 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 3\nu^{2} + 2\nu + 1$$ v^4 - 2*v^3 - 3*v^2 + 2*v + 1 $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 2\nu^{3} + 4\nu^{2} - 4\nu - 2$$ -v^4 + 2*v^3 + 4*v^2 - 4*v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2\beta _1 + 1$$ b4 + b3 + 2*b1 + 1 $$\nu^{3}$$ $$=$$ $$2\beta_{4} + 2\beta_{3} + \beta_{2} + 7\beta_1$$ 2*b4 + 2*b3 + b2 + 7*b1 $$\nu^{4}$$ $$=$$ $$7\beta_{4} + 8\beta_{3} + 2\beta_{2} + 18\beta _1 + 2$$ 7*b4 + 8*b3 + 2*b2 + 18*b1 + 2

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.79802 1.17316 0.266708 −0.787589 −1.45030
−1.79802 1.59027 1.23287 0 −2.85933 1.00000 1.37931 −0.471046 0
1.2 −0.173158 −1.11762 −1.97002 0 0.193524 1.00000 0.687440 −1.75093 0
1.3 0.733292 2.28713 −1.46228 0 1.67714 1.00000 −2.53886 2.23098 0
1.4 1.78759 −0.0742225 1.19548 0 −0.132679 1.00000 −1.43816 −2.99449 0
1.5 2.45030 3.31444 4.00395 0 8.12135 1.00000 4.91027 7.98549 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4025.2.a.r 5
5.b even 2 1 805.2.a.k 5
15.d odd 2 1 7245.2.a.bi 5
35.c odd 2 1 5635.2.a.x 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.k 5 5.b even 2 1
4025.2.a.r 5 1.a even 1 1 trivial
5635.2.a.x 5 35.c odd 2 1
7245.2.a.bi 5 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{5} - 3T_{2}^{4} - 2T_{2}^{3} + 10T_{2}^{2} - 4T_{2} - 1$$ T2^5 - 3*T2^4 - 2*T2^3 + 10*T2^2 - 4*T2 - 1 $$T_{3}^{5} - 6T_{3}^{4} + 8T_{3}^{3} + 7T_{3}^{2} - 13T_{3} - 1$$ T3^5 - 6*T3^4 + 8*T3^3 + 7*T3^2 - 13*T3 - 1 $$T_{11}^{5} + 11T_{11}^{4} + 14T_{11}^{3} - 185T_{11}^{2} - 612T_{11} - 452$$ T11^5 + 11*T11^4 + 14*T11^3 - 185*T11^2 - 612*T11 - 452

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 3 T^{4} - 2 T^{3} + 10 T^{2} + \cdots - 1$$
$3$ $$T^{5} - 6 T^{4} + 8 T^{3} + 7 T^{2} + \cdots - 1$$
$5$ $$T^{5}$$
$7$ $$(T - 1)^{5}$$
$11$ $$T^{5} + 11 T^{4} + 14 T^{3} + \cdots - 452$$
$13$ $$T^{5} - 7 T^{4} - 15 T^{3} + 146 T^{2} + \cdots - 761$$
$17$ $$T^{5} + T^{4} - 60 T^{3} + 15 T^{2} + \cdots - 1444$$
$19$ $$T^{5} - 2 T^{4} - 45 T^{3} + 153 T^{2} + \cdots - 452$$
$23$ $$(T + 1)^{5}$$
$29$ $$T^{5} + 14 T^{4} - 39 T^{3} + \cdots - 7448$$
$31$ $$T^{5} + 6 T^{4} - 70 T^{3} + \cdots + 2489$$
$37$ $$T^{5} - T^{4} - 168 T^{3} + \cdots - 13628$$
$41$ $$T^{5} - 7 T^{4} - 70 T^{3} + 146 T^{2} + \cdots - 331$$
$43$ $$T^{5} - 12 T^{4} - 77 T^{3} + \cdots - 8236$$
$47$ $$T^{5} - 24 T^{4} + 196 T^{3} + \cdots + 289$$
$53$ $$T^{5} - 21 T^{4} + 142 T^{3} + \cdots + 548$$
$59$ $$T^{5} + T^{4} - 70 T^{3} - 131 T^{2} + \cdots + 176$$
$61$ $$T^{5} - 7 T^{4} - 46 T^{3} + \cdots - 2252$$
$67$ $$T^{5} - 35 T^{4} + 450 T^{3} + \cdots + 596$$
$71$ $$T^{5} + 32 T^{4} + 264 T^{3} + \cdots - 54361$$
$73$ $$T^{5} - 7 T^{4} - 89 T^{3} + \cdots + 4297$$
$79$ $$T^{5} - 18 T^{4} - 19 T^{3} + \cdots - 11236$$
$83$ $$T^{5} - T^{4} - 204 T^{3} + \cdots + 10204$$
$89$ $$T^{5} - T^{4} - 230 T^{3} + \cdots - 16204$$
$97$ $$T^{5} - 9 T^{4} - 271 T^{3} + \cdots - 4948$$